1. A Topological Extension of GR: Black Holes Induce Dark Energy Marco Spaans University of Groningen spaans@astro.rug.nl

2. General Relativity Equivalence principle: Einstein’s local interpretation of Mach’s global principle. GR not invariant under local conformal transformations → singular metric fluctuations on the Planck scale: Wheeler’s (1957) quantum foam of wormholes. GR elegant and succesful, preserve in quantum gravity. The Einstein equation G μν = 8πG N T μν does not specify local and global topology.

3. The Equivalence Principle: Macro- versus Microscopic paths Need to specify what path to take quantum- mechanically.

4. Put Differently: The Feynman (1948) Path Integral A particle/wave travels along many paths as an expression of the superposition principle: connect any A to any B.. ∫ paths e iS leads to a semi-classical world line and a large scale limit for GR. But, how to distinguish paths on L Planck ? Observers cannot tell paths apart locally due to errors! This talk: Choose paths that can be identified despite quantum uncertainty in geometry; paths fundamental and rooted in topology ; Spaans 1997, Nuc.Phys.B 492, 526.

5. ↔ Space-time on the Planck scale (10 -33 cm) is a fabric woven of many paths: shape → connectivity.

6. Physics: Multiplicity Scrutinize down to Planck scale: strong coupling between observer and observee. To identify a path, which is distinct under continuous deformation of space-time, one needs a proper example to compare to. Thought: It takes one to know one. So topological fluctuations on L Planck take the form of multiple copies of any path through quantum space-time (i.e., in 4D). Idea: Topological dynamics on L Planck because no distinct path can (remain to) exist individually under the observational act of counting, thus building up quantum space-time.

7. Mathematics: Loop Algebra Use 3D prime manifolds P: T 3, S 1 xS 2, S 3 to construct a topological manifold, with the S 1 loop as central object: Two distinct paths from A to B together form a loop. T 3, three-torus (Χ=0→LM), provides topologically distinct paths through space-time. Superposition by Feynman. S 1 xS 2, handle (LM), can accrete mass as well as Hawking (1975) evaporate (so BH~wormhole in 4-space). The handle represents Wheeler’s quantum foam. S 3, unit element (simply connected, large scale limit). Minimal set. Physical principle: find the paths that thread Planckian space-time (Feynman+Wheeler). Mathematical principle: topological constructability (primes+homeomorphisms) ; Spaans 2013, J.Phys 410, 012149

8. Constructing Space-Time Topology Define Operators T (annihilation) and T* (creation): TP ≡ nQ, n=number of S 1 loops and Q is P with an S 1 contracted to a point (dimension-1). T*P ≡ PxS 1 (dimension+1). [T,T*] = 1 (non-commuting). NS 3 = S 3, NT 3 = 7T 3, NS 1 xS 2 = 3S 1 xS 2. N≡T*T+TT* (symmetric under T ↔ T*): counting. Repeated operations with N allow one to build a space- time with Planck scale heptaplets of three-tori and Planck mass triplets of handles embedded in 4-space.

9. Equation of Motion: Counting = Changing n[T 3 ](m) = 7 m for discrete m=0,1,2,…; topological time t=(m+1)T Planck and a continuous interval (m+) ≡ (m,m+1). n[S 1 xS 2 ](m+1) = 3n[S 1 xS 2 ](m) +Form(m+) -Evap(m+) -Merg(m+) → n[S 1 xS 2 ](m) = 3 m if F,E,M small (drives inflation), 2n[S 1 xS 2 ](m) = Evap(m+) if F,M and δn[S 1 xS 2 ] small; for every Planckian volume, independent of BH location. Matter degrees of freedom F,E,M modify the number of handles: GR part of quantum space-time through longevity and transience of all BHs.

10. Topologically distinct paths, under homeomorphisms, between any two points: Use T 3, S 1 xS 2 and S 3 Build a lattice of three-tori, with spacing L Planck, to travel through 4D space-time as an expression of the superposition principle and attach Wheeler’s handles: Quantum paths that connect points are then globally identified through non-contractible loops. Multiply Connected Space-Time

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12. Topological Induction of BHs Solution 2θ(m) = Evap(m+), for the number θ of BHs in a time slice of width ~T Planck : Space-time responds globally to local BHs by inducing Planckian BHs. Wave function of the universe collapses when BH forms. Planckian BHs evaporate in about T ev ~T Planck. Macroscopic BHs θ living longer than a Hubble time generate a globally stable quantum foam density Λ. Extension of Einstein gravity and Mach’s principle: Global topology and geometry determine (changes in) Planck scale topology and the motions of matter, and vice versa. Λ = θ m Planck / L f 3, with L f the size of the universe when the first long-lived BH, so with T ev [m']>T univ [m], forms: L f is frozen in when one has a global 4-space topology in the sense of Mach and the quantum foam stabilizes.

13. Dark Energy Cosmological observations suggest an accelerated expansion of the universe, usually attributed to some form of vacuum energy Λ: Einstein’s cosmological constant in the form of dark energy (Riess ea 98, Perlmutter ea 98, etc.), with ρ 0 ~10 -29 g cm -3. This while θ~10 19 today. Topological induction then requires the dark energy to follow the number of macroscopic BHs in the universe. Induced Planckian BHs require an increase in 4-volume for their embedding: L f ≈2x10 14 cm from ρ 0 and θ. Observed star/BH formation history of universe yields: Λ(z)/Λ(0)~(1+z) -0.36, z<1; a ~30% decline: w ≈ -1.1, not constant, consistent with observations (Aykutalp & Spaans 12). In fact, w = -1.13 ± 10% (Hinshaw ea 12; Planck: Ade ea 13).

15. Summary Use the topological freedom in GR to implement the superposition principle through the notion of globally distinct paths for particles to travel along. The S 1 loop dynamics of three-tori and handles generate such paths: multiply to identify. Leads to the creation of Wheeler’s Planckian BHs by macroscopic BHs, i.e., a testable form of dark energy. What if (many) primordial BHs evaporate until today? Inflation driven by induced mini BHs: n≥55 e-foldings, n~ln(L f /L Planck ) 1/2 ~lnθ 1/6. Further scrutiny of the lattice of three-tori is necessary: fluctuations with n s =1-r ≈ ln(7)/n = 0.96 and macroscopic temporal displacement in the CMB with amplitude 1/14.